Notable Properties of Specific Numbers

15 is the magic constant for a 3×3 magic square, the smallest
possible nontrivial magic square. An N×N magic square consists
of the N2 integers from 1 to N arranged in an N×N square
grid, such that the sum of any row, or any column, or of one of the
two diagonals, is equal to any other. There is only one solution for a
3×3 square, not counting its rotations and reflections:

4 9 2
3 5 7
8 1 6

This magic square was known to the Chinese at least 3000 years ago and
is called the Lo Shu. It appears in legends and artwork; for an
example see here.

15 is the sum of the first 5 numbers: 1+2+3+4+5=15. Such numbers are
called triangular because 15 things can be arranged in a triangular
shape, by putting 1 object in a top "row", 2 in the 2nd row, and so
on down to 5 in the 5th row. The general formula for the Nth
triangular number is N(N+1)/2. Because either N or N+1 is
even, we know that either N/2 or (N+1)/2 is an integer. It follows
that every triangular number is the product of two integers  either
N × (N+1)/2 or N/2 × (N+1)  and therefore, every
triangular number above 3 is composite.

The sum 1+2+3+4+5=15 is a little more significant because the first
and last numbers, 1 and 5, are also the digits of the total. See
27.

Base 16, called hexadecimal, is the most popular base among
computer programmers for representing raw data in computer memory. The
"digits" A, B, C, D, E and F are used to represent values
of 10 through 15. So, for example, the hexadecimal number 6A16 is
6×16+10 = 106.

Until the mid-1970's, base 8 was the most common base in
computer programming applications. The primary reason base 16 overtook
base 8 is that it uses 4 bits per digit and 4 itself is a power of 2.
All the popular microprocessors, from the very early ones in the
1970's, have been based on a power of 2 bits per machine word.

Peter, a film critic, enjoys going to eclectic cocktail parties to
meet the directors and producers. He has noticed that for any two
people in a party, they have always either A) met each other at the
studio, B) met each other at the Academy Awards, or C) met each other
here because they haven't met before the party. Peter got to
wondering, how many people would need to be at a party to guarantee
that there is at least one group of three people who have met each
other in the same place (A B or C)? The answer is 17.

17 is the smallest number that can be written as A2 + B3 in
two different ways: 17 = 32 + 23 = 42 + 13. By the way,
the pair (8, 9) is the only pair of consecutive numbers where one is a
square and the other is a cube (Euler proved this.)
Catalan's conjecture (proven in 2002) states that 23 and 32 is the
only pair of consecutive powers, aside from trivial cases where one of
the numbers is a "1st power".

Any convex polyhedron has at least one face that it can rest on
without falling over (proof: if it didn't, it would be a
perpetual-motion machine!). Most have more than one stable face. The
minimum number of faces on a polyhedron that has only one stable
face is 17. (The assumption is that the polyhedron is rigid, solid
(not hollow) and of uniform density.)

The reciprocal of 17, 1/17=0.05882352941176470588235..., has a
16-digit repeating decimal, which is the longest possible. This is a
property that 17 shares with 7 and with many higher prime
numbers. It results from the fact that, when you perform long division
to compute 1/17, every remainder except 0 comes up exactly once. Also
because of this, the multiples 2/17, 3/17, and so on have decimal
fractions that use the same set of 16 digits, but starting in a
different place (just as seen with 1/7, 2/7, 3/7 etc.). A reader named
Jeremy pointed out to me that these 16 digits reduce to 9 using the
casting out 9's technique, and thus 0588235294117647 is a
multiple of 9. This holds for all reciprocals of primes (starting with
7) and is a consequence of Fermat's little theorem.

These are two closely-related types of numbers. 17 belongs to both
classes.

A "cult" number has a "following", a group of "fans", many of whom
have set up web pages for the number, a sort of virtual shrine to the
number. Typically, a cult-number fan is someone who has one favorite
number, and who delights in noticing that number, whenever it occurs
in a place that seems to be more than just coincidence. I seem to have
chosen 27, because I notice that number a lot more
than I "should".

Other cult numbers include 23, 37, 42,
47, 69, and 666; closely related to these are
numbers that attract an abnormal amount of quasi-scientific interest,
such as the fine structure constant.

A "psychologically random" number, is one that "sounds random", or
is chosen more often when someone is asked to pick a random number. 17
is the most often picked number in response to the request "Pick a
random number from 1 to 20." Psychologically random numbers are used
by writers when some large number is needed but no particular value is
better than any other. For example, see 37 for some examples
from movies.

Psychologically random numbers are usually odd and don't end in 5,
because there is a natural psychological bias to thinking even numbers
and numbers that end in 5 are "less random". This means that most
psychologically random numbers are prime. An extended argument along
the same lines (attributed to Hilary Putnam by Mark Kalderon), adds
that 7, 11 and 13 are nonrandom because of their lucky and unlucky
associations, 9 is a square and 3 is "for the Trinity", leaving 17 as
the only number less than 20 that isn't special.

When a number is not consciously chosen but just happens by
accident, it is more likely to be noticed and perceived as "more than
just a coincidence" if the number is psychologically random. When such
numbers are noticed repeatedly, they can then become
cult numbers. This is why many cult numbers are also
psychologically-random numbers.

18 years is the amount of time it takes for the tilt of the Moon's
orbit to rotate a full-circle. Another way of saying the same thing is
that the eclipse season slips back an entire year
 so every 18 years (more precisely, 18 years 11 days), there have
been 38=2×19 eclipse seasons.

Viewed simply as a close match between a multiple of years (18) and
eclipse seasons (38), it isn't such a big deal. It's 11 days off, and
if you wait a year, you actually get a closer match (40 eclipse
seasons is 8 days shorter than 19 tropical years). What
makes the 18-year period so special is that the number of
synodic months involved (223) is only 0.04 days short
of 242 draconic months and only 0.2 days short of 239
anomalistic months. This means that the eclipses, in
addition to being at the same time of year, also have the same
positioning of Moon and Earth in the other two dimensions (distance
and north-south positioning). On each eclipse, the Moon is the same
distance away from Earth, and in the same position north-to-south, as
it way 18 years 11 days previously. The distance governs whether solar
eclipses are annular or total, and the north-south positioning
determines where the shadow crosses for both types of eclipses (lunar
and solar).

This is the saros, an incredibly rare coincidence that makes
eclipses easy to predict. It was discovered by the Babylonians and the
knowledge was passed on to the Greeks and thence to later
civilizations.

The coincidence is not exact; each saros the Moon is a little
further north (on a descending node) or south (on an ascending node)
than the previous time, by a distance about equal to 1/70 of the
Earth's diameter. So, each repeated eclipse repeats about 70 times, or
about 70×18=1260 years.

Also, the saros period is
not an exact number of days; the Earth has turned
about 1/3 of the way around, so the eclipses (particularly solar) are
not seen by the same people.

19 is the number of years in the repeating pattern of a lunisolar
calendar (such as the Hebrew calendar) designed to use months that
stay in phase with the moon while also having the year stay in phase
with the seasons. 19 tropical years is 6939.60160373
mean solar days, only 2 hours 5 minutes shorter than 235
synodic months (which is 6939.68838046 mean solar
days). Due to this rather happy coincidence, the phases of the moon
fall on the same dates every 19 years. It takes over 200 years for the
error to get to be more than a day (however, in order to accomplish
this accuracy, the lengths of at least one of the 235 months must
differ from one 19-year cycle to another, see 6940). This
cycle was known to several cultures at least as far back as the 4th
century BCE. The period of 19 years also figures in the calculation of
the date of Easter (but this is complicated by an additional multiple
of 7 due to the fact that Easter must fall on a Sunday, see
133).

Due to another amazing coincidence, 19 tropical years is also
(within less than a day) equal to 255 draconic months, which means
that eclipses also repeat every 19 years. (However, the pattern only
repeats 4 or 5 times before an eclipse stops happening on a given
date; the saros is a much better match)

19 is the numerical value of the Arabic word
wahid ("one"), one of the names of God. 19 is considered sacred by
the Baha'is, an Islamic sect, and they divided the year up into 19
"months" of 19 days each (this makes 192=361 days; to round out the
year an intercalary period of 4 or 5 days is added)9. They also
group years into 19-year and 361-year cycles (see Bahai calendar).

Someone wrote to me pointing out that 19 has a "special" property:
It is the sum of 9 and 10, and also the difference of their
squares: 100 - 81 = 19. Of course, this property is true for
any number that can be expressed in the form X + X + 1, which
means pretty nearly any number (to prove, expand (X+1)2 and then
subtract X2.). This is an example of the type of property that is
often reported for cult numbers like 23 and
psychologically random numbers like 37,
because it is often desirable to find as many properties of such
numbers as possible.